prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$

Let $q\in \mathbb{Z}$ be a prime greater than $2$, let ${\zeta }_q=e^{2\pi i/q}$ and write $L=\mathbb{Q}({\zeta }_q)$ for the cyclotomic extension.The ring of integers of $L$ is ${𝒪}_L=\mathbb{Z}[{\zeta }_q]$. Thediscriminant of $L/\mathbb{Q}$ is:

$$D_{L/\mathbb{Q}}=\pm q^{q-2}$$

and it is $+$ exactly when $q-1\equiv 0,1\mathrm{mod}\mathrm{\hspace{0.25em}4}$.

Proposition 1.

$\sqrt{\pm q}\in \mathbb{Q}({\zeta }_q)$, with $\mathrm{+}$ exactly when $q\mathrm{-}\mathrm{1}\mathrm{\equiv }\mathrm{0}\mathrm{,}\mathrm{1}\mathit{}\mathrm{mod}\mathit{}\mathrm{\hspace{0.25em}4}$.

Proof.

It can be proved that:

Taking square roots we obtain

Hence the resultholds (and the sign depends on whether $q-1\equiv 0,1\mathrm{mod}\mathrm{\hspace{0.25em}4}$).∎

Let $K=\mathbb{Q}(\sqrt{\pm q})$ with the corresponding sign. Thus, bythe proposition we have a tower of fields:

For a prime ideal $p\mathbb{Z}$ the decomposition in the quadraticextension $K/\mathbb{Q}$ is well-known (see http://planetmath.org/encyclopedia/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ.htmlthis entry). The next theoremcharacterizes the decomposition in the extension $L/\mathbb{Q}$:

Theorem 1.

Let $p\mathrm{\in }\mathrm{Z}$ be a prime.

1. If $p=q$, $q{𝒪}_L={\left(1-{\zeta }_q\right)}^{q-1}$. Inother words, the prime $q$ is totally ramified in $L$.

2. If $p\ne q$ then $p\mathbb{Z}$ splits into $(q-1)/f$ distinctprimes in ${𝒪}_L$, where $f$ is the order of $p\mathrm{mod}q$ (i.e. $p^f\equiv 1\mathrm{mod}q$,and for all ).